We prove necessary and sufficient Tauberian conditions for locally integrable functions (in Lebesgue's sense) over R + , under which convergence follows from summability by weighted mean methods. The main results of this paper apply to all weighted mean methods and unify the results known in the literature for particular methods. Among others, the conditions in our theorems are easy consequences of the slow decrease condition for real-valued functions, or the slow oscillation condition for

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... ex-valued functions. Therefore, practically all classical one-sided as well as two-sided Tauberian conditions for weighted mean methods are corollaries of our two main theorems. Summability of integrals over R + by weighted mean methods Let P be a function defined on R + := [0, ∞) such that P is nondecreasing on R + , P(0) = 0 and lim t→∞ P is called a weight function, due to the fact that it induces a positive Borel measure on R + .